Knot Concordance and Metric Spaces

William Marsh Rice University, Houston TX

Investigators

Abstract

This research project concerns low-dimensional topology, which studies the properties of spaces called manifolds that locally look like the world we live in. Many of the mathematical tools successfully used to study high-dimensional manifolds do not apply to manifolds in four or fewer dimensions, and important fundamental problems remain unresolved. This project studies knots and links, knotted circular strings in three-dimensional space. Three-dimensional manifolds can be mathematically represented by framed links, where the so-called framings are instructions to modify the space near the link. As a consequence, the study of knots and links is a fundamental tool in understanding three-dimensional manifolds. In this project, the investigator aims to get a better understanding of how knots behave as they move through time as a fourth dimension. This line of investigation constitutes an important subfield of low-dimensional topology known as concordance. An algebraic operation with properties similar to the addition of numbers can be performed on the collection of knots, and this leads to the construction of an algebraic object known as the concordance group. The concordance group is an extremely complicated entity, which is poorly understood despite great efforts by many mathematicians. To gain perspective on its structure, the investigator will introduce a notion of distance and investigate how certain geometric operations change this distance. The primary goal of this project is to gain a better understanding of knot and link concordance. The investigator will do this by studying the concordance group as both an abelian group with associated filtrations, such as the n-solvable and bipolar filtrations, and also as a metric space with discrete and non-discrete metrics. Specific projects are as follows: (1) show that the "other half'" of the n-solvable filtration is nontrivial; (2) determine if successive quotients of the n-solvable filtration of the link concordance group are abelian; (3) understand the relationship between the Milnor invariants of derivatives of a knot and the concordance class of the knot; (4) view the smooth knot concordance group as a metric space and study natural geometric operators acting on it (in particular, show that there is a non-discrete metric on the subgroup of topologically slice knots T and, in addition, show that the successive quotients of the bipolar filtration are nontrivial on T); (5) define the knot complex and study its algebraic topology.